A topological space $(X,\tau)$ is said to be homogeneous if for any $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$. Moreover, we say that $(X,\tau)$ is shrinkable if there is $S\subseteq X$ with $S\neq X$ and a homeomorphism $\psi: X\to S$ where $S$ is endowed with the subspace topology.

The real intervai $[0,1]$ is shrinkable (it is homeomorphic to $[0,1/2]$), but not homogeneous. $S^1$ is homogeneous, but not shrinkable.

Is there an example of a shrinkable, homogeneous, connected, and compact $T_2$-space?

A topological space $(X,\tau)$ is said to be homogeneous if for any $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$. Moreover, we say that $(X,\tau)$ is shrinkable if there is $S\subseteq X$ with $S\neq X$ and a homeomorphism $\psi: X\to S$ where $S$ is endowed with the subspace topology.

The real interval $[0,1]$ is shrinkable (it is homeomorphic to $[0,1/2]$), but not homogeneous. $S^1$ is homogeneous, but not shrinkable.

Is there an example of a shrinkable, homogeneous, connected, and compact $T_2$-space?